There are just two points between Inter and third-placed AC Milan, who could go top by beating Sampdoria on Sunday.
Napoli, who have played a game more than Inter, took an early lead when Lorenzo Insigne slotted in a penalty.
Dzeko equalised for Inter, who play Liverpool in the Champions League on Wednesday, early in the second half.
Napoli created the better openings as Piotr Zielinski hit the post early in the first half before Inter goalkeeper Samir Handanovic was forced into a smart save to deny Elif Elmas later on.
Insigne’s goal meant he surpassed Diego Maradona (115) to become Napoli’s third-highest goalscorer in all competitions. Only current players Dries Mertens (144) and Marek Hamsik (121) have more goals for the club.